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SEMI ME1392-0305 © SEMI 2003, 2005 13 R1-3 Reflectance Factor R1-3.1 A measure of diffuse reflectance in common use is the reflectance factor, R , that is the ratio of flux propagated from source to r eceiver in a reflec…
SEMI ME1392-0305 © SEMI 2003, 2005 12
RELATED INFORMATION 1
ADDITIONAL DATA PRESENTATION
NOTICE: This related information is not an official part of SEMI ME1392 and is not intended to modify or
supercede the officiall standard. It was developed during the original approval of this standard by ASTM
Committee E12 in 1996. SEMI approval was by full letter ballot procedures with publication authorized by the NA
Regional Standards Committee on December 10, 2004. Determination of the suitability of the material is solely the
responsibility of the user.
R1-1
Presentation
R1-1.1 It is common practice to plot BRDF with respect to the angle from the specular beam,
. If scatter is
measured only in the PLIN,
=
s
i
. However, in the more general case for scatter out of the PLIN:
ssisi
cossinsincoscoscos
1
(R1-1)
R1-1.2 This is a useful angular reference for specular samples. However, when using this format, care must be
taken that
is not confused with
s
in the calculation of BRDF. This presentation format is normally used only
when
passes through zero, that is, when the scatter scan includes the specular beam.
R1-1.3 The terms “forward scatter” and “back scatter” refer to PLIN scatter directions for which
is respectively
positive or negative. Note that
continues to increase as a negative angle when passing the surface normal since
the sign of
s
switches in the above equation.
R1-2
Presentation
R1-2.1
=
o
, where
= sin
s
and
o
= sin
i
, is a method of expressing the angle between the specular and
scatter directions in direction cosine space along the surface for scatter in the PLIN. This is a very useful
normalization when scatter results only from surface microroughness, and the grating equation:
fn
is
sinsin (R1-2)
where:
= wavelength of the incident flux,
f = linear spatial frequency for the microroughness in the x or y direction, and
n = diffraction order.
This equation can be used to relate
s
to the frequency, f. Only the first order (n = 1) is significant for roughness
much less than the wavelength. BRDF can now be interpreted as the ability of each frequency to scatter light. If
BRDF is plotted against a
scale it may be independent of
i
and proportional to f. If the surface behaves in this
way the BRDF is “shift invariant.”
11
R1-2.2 In the general case for scatter out of the PLIN the following two dimensional grating equations apply:
xiss
fn
sinsincos (R1-3)
and
yss
fn
sinsin
(R1-4)
The definition of
must be expanded to include the projection of the scattered light in the X and Y directions:
ssisi
cossinsin2sinsin
222
(R1-5)
11 Harvey, James E., “Light Scattering Characteristics of Optical Surfaces,” Proceedings SPIE 107, 41 (1977)
SEMI ME1392-0305 © SEMI 2003, 2005 13
R1-3 Reflectance Factor
R1-3.1 A measure of diffuse reflectance in common use is the reflectance factor, R, that is the ratio of flux
propagated from source to receiver in a reflectometer with a specimen, to the flux propagated with a perfectly
reflecting diffuser. Regarding a scatterometer as a very directional bi-directional reflectometer, the following
relationship between R and BRDF is obtained as follows:
BRDF
1/
BRDF
BRDF
BRDF
diffuser
R
(R1-6)
Additional information can be found in ASTM Practice E 167. Note that reflectance factor and specular reflectance
share the same symbol, R, but they are not the same parameter.
SEMI ME1392-0305 © SEMI 2003, 2005 14
RELATED INFORMATION 2
MODEL DEPENDENT CALCULATED PARAMETERS
NOTICE: This related information is not an official part of SEMI ME1392 and is not intended to modify or
supercede the official standard. It was developed during the original approval of this standard by ASTM Committee
E12 in 1996. SEMI approval was by full letter ballot procedures with publication authorized by the NA Regional
Standards Committee on December 10, 2004. Determination of the suitability of the material is solely the
responsibility of the user.
R2-1 Total Integrated Scatter (TIS)
R2-1.1 TIS can be calculated from BRDF by integrating BRDF over the hemisphere.
12
Typically a 5° total angle
“hole” is left around the specular beam since specular light is not included in total integrated scatter (see SEMI
MF1048).
R2-1.1.1 For an isotropic surface, measure in-plane BRDF at
i
= 0 and calculate the expected total integrated
scatter by integrating over the angle limits specified in SEMI MF1048.
70
5.2
1
calculated
dsinBRDFcos2TIS
sss
R
(R2-1)
R2-1.1.2 Sample specular reflectance, R, must be included because total integrated scatter is referenced to reflected
and not incident power. The cos
s
term must be included because BRDF is defined in terms of the projected
receiver aperture. This comparison between total integrated scatter and BRDF may not be exact since the total
integrated scatter detector is less sensitive to light incident on the detector at large angles and if low f (close to
specular) scatter dominates, the 5° hole size is critical. In addition a TIS instrument is not polarization selective.
R2-2 Roughness
R2-2.1 The rms surface roughness,
, is an often quoted number that can be obtained from direct profile
measurements with stylus or optical profilometers. It can also be inferred from total integrated scatter when,
<<
/4
for front surface scatter from a clean, smooth surface,
13
as described in SEMI MF1048.
2/1
TIS
4
(R2-2)
R2-2.1.1 The user must confirm the usefulness of this
calculation based on the particular measurement
circumstances. It may have strong frequency limitations and not agree with surface roughness derived from optical
or mechanical profile instruments (which can have different spatial frequency limits
14
).
R2-3 Power Spectrum
R2-3.1 The surface power spectral density function (PSD) can be calculated from the BRDF through a scatter
model. For example, the grating equation model discussed in §R1-2 shows that high frequency surface
perturbations scatter light far from specular and low frequency perturbations scatter close to specular. The PSD
shows the amount of modulation versus f , that is, the square of the Fourier transform of the surface profile. Since it
is a sample property, the same PSD should be obtained regardless of wavelength and incident angle dependent
differences in the BRDF data.
R2-3.2 Wavelength scaling is another check on system calibration. Smooth, clean, nonabsorbing front surface
reflectors should yield the same PSD for different BRDF measurement wavelengths. If the instrument does not
wavelength scale on appropriate samples, the BRDF measurement may be suspect. Polished molybdenum and
12 Stover, John C., Hourmand, Bahram, and Kahler, Jeffrey, A., “Comparison of Roughness Measurements by Differential Scatter and Total
Integrated Scatter,” Proceedings SPIE 511, 2-6 (1984).
13 Stover, John C., Optical Scattering: Measurement and Analysis, 2nd Edition, (SPIE Optical Engineering Press. Bellingham, WA, 1995)
Chapter 4.
14 Bennet, Jean M., and Mattsson, Lars, Introduction to Surface Roughness and Scattering (Optical Society of America, Washington, DC, 1989)
p. 32.